Table of contents

Asymmetric Cyptography

Every participant has a key pair. The key pair consist of:

• public key
• private key

Math

Take RSA.

1. Pick two large primes, p & q.
2. Compute their product, n.
3. Compute Euler’s totient $\phi(n) = (p - 1)(q - 1)$
4. Pick a public exponent e so that it causes wrap around with the modulus.
5. Compute the private exponent d $d \cdot e \equiv 1 \pmod{\phi(n)}$

Digital Signature

You have a message. You hash it using a known hashing algorithm. Sign the hash with your private key. At this point, you have:

• M = the message
• H = hash(M), the hash of the message body
• S = encrypt_private(H), the signed hash with private key

The reciever can then:

• H = hash(M) <- reciever has to do this step; both sides have to do it
• $H_s$ = decrypt_public(S)

The reciever takes the signed hash and can verify it with the public key to get back the original hash.